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Unformatted text preview: Solutions to Problem Set 8 Econ 202A  2nd Half  Fall 2011 Prof. David Romer, GSI: Victoria Vanasco 1 Consumption and Risky Assets Consumer's lifetime utility: U = u ( c 1 ) + E [ u ( c 2 )] Income: Y 1 = ¯ Y certain and Y 2 ∼ F ( ¯ Y ,σ 2 y ) is random variable. Initial wealth A is zero. a. With no nancial assets and a storing technology available, the problem is given by: U = u ( c 1 ) + E [ u ( c 2 )] c 1 + s = ¯ Y c 2 = Y 2 + s FOC wrt s (by choosing how much to save the consumer pins down ( c 1 ,c 2 ) : u ′ ( c 1 ) = E [ u ′ ( c 2 )] Now, we add a risky asset to this economy, with return r ∼ ( ¯ r,σ 2 r ) , with E ( rY 2 ) = 0 . b. The consumer will buy a strictly positive amount i ¯ r > , i.e., if the expected return on this investment exceeds the risk free rate (in our case zero), it will be optimal for the consumer to invest at least an ε > amount of its income in the risky asset. This is true because for a small ε , consumption is period 2 is "almost" uncorrelated with the return on this asset (since C 2 = Y 2 + ε (1 + r ) ), so the covariance is zero and it has positive expected return ⇒ the consumer should invest some positive amount. Also, notice that for very small amounts of risk, the utility function can be aproximated by a linear function (think of a taylor expansion around Y 2 ), i.e. agents are 'almost' risk neutral when it comes to small amounts of risk. This are both reasons why the agent always takes some risk (as long as the risk is small, the linear aproximation works). c. This statement is not true because the consumer is risk averse ( u ′′ < 0) , and thus won't be willing to leverage up to in nity to go long on the risky asset. There is an optimal level (interior solution) that will determine the optimal amount to be invested in the risky asset. The di erence with out answer to part b. is that for not small amounts of investment in the risky asset, consumption and the return on the asset become highly correlated, and thus the asset becomes less and less attractive since remember the agent ultimately cares about correlation between returns and marginal utility of consumption! 1 d. Consumer now has to choose how much to invest in the risk free and how much to invest in risky asset. Let α be the fraction of income invested in risky asset and β the fraction invested in risk free. The consumer needs to choose α and β to max expected utility: max { α,β } u ( c 1 ) + E [ u ( c 2 )] c 1 = ¯ Y (1 α β ) c 2 = Y 2 + [(1 + r ) α + β ] ¯ Y FOC ( α ) u ′ ( c 1 ) ¯ Y + E [ u ′ ( c 2 ) (1 + r )] ¯ Y = ( β ) u ′ ( c 1 ) ¯ Y + E [ u ′ ( c 2 )] ¯ Y = Can be rewritten as follows: u ′ ( c 1 ) = E [ u ′ ( c 2 ) (1 + r )] (1) u ′ ( c 1 ) = E [ u ′ ( c 2 )] (2) Where (1) is the Euler equation for the risky asset and (2) the Euler equation for the riskfree asset (equal to the ne derived in part a.) 2 The Risk Free Rate Puzzle We solved this in section a long time ago! The quickest way is to use the result:We solved this in section a long time ago!...
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 Fall '07
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 Utility, Prof. David Romer

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